Are old aircraft the only ones approved for zero-g?
Not at all. Its simply that older airliners close to the end of their airframe hours limit are VASTLY cheaper to buy and then convert for that purpose. As an airliner, the remaining airframe hours would get used up far too quickly, but for zero-G training those hours can last many more years with the typical usage profiles of such aeroplanes. A lot of the removed interior fittings can also be resold to slightly offset the cost.
Heck no. Any airliner has to be certified to handle negative g forces, typically at least -2 g, to cater for events like flying through heavy turbulence. By comparison, zero g is rather benign. You see old airlines being used for such purpose because they are a lot cheaper than newer planes to buy, and since they don't go far, they do not need to be as fuel efficient flying essentially empty of payload.
No. Technically, any aircraft can be approved for zero-G, it's just that the acquisition costs for an older aircraft is a small fraction of a new one. Typically, a plane in airline service flies 9 to 12 hours per day. A zero-G plane flies maybe 10-20 hours per week. With so much downtime you don't want to have a $60 to $150 million asset sitting idle 150 hours per week. On the other hand, the revenue you get from using the plane for zero-G flights a couple of days a week may cover the cost of a $5 million aircraft.
Any aircraft can do zero gravity, it is called physics.